The net force, which is the centripetal force in this case, goes towards the centre of the circular path of rotation.
If you think about it, if the net force was towards the centre of the chain, the person in the device would accelerate upwards, towards the centre of the chain, but they clearly instead follow a circular path.
We find the magnitude of the tension by realising that the net force must be the centripetal force, and so the vertical component of the tension must negate the person's weight force, while the horizontal component of the tension provides the centripetal force. With these two figures, we can calculate the magnitude of the tension in the chain with Pythagoras' theorem.